ARTICLE
Received 13 Oct 2014 | Accepted 21 Apr 2015 | Published 8 Jul 2015
DOI: 10.1038/ncomms8257
Constant-intensity waves and their modulation
instability in non-Hermitian potentials
K.G. Makris1,2, Z.H. Musslimani3, D.N. Christodoulides4 & S. Rotter1
In all of the diverse areas of science where waves play an important role, one of the most
fundamental solutions of the corresponding wave equation is a stationary wave with constant
intensity. The most familiar example is that of a plane wave propagating in free space. In the
presence of any Hermitian potential, a wave’s constant intensity is, however, immediately
destroyed due to scattering. Here we show that this fundamental restriction is conveniently
lifted when working with non-Hermitian potentials. In particular, we present a whole class
of waves that have constant intensity in the presence of linear as well as of nonlinear
inhomogeneous media with gain and loss. These solutions allow us to study the fundamental
phenomenon of modulation instability in an inhomogeneous environment. Our results pose a
new challenge for the experiments on non-Hermitian scattering that have recently been put
forward.
1 Institute for Theoretical Physics, Vienna University of Technology, Vienna A-1040, Austria. 2 Department of Electrical Engineering, Princeton University,
Princeton, New Jersey 08544 USA. 3 Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510, USA. 4 College of
Optics–CREOL, University of Central Florida, Orlando, Florida 32816, USA. Correspondence and requests for materials should be addressed to K.G.M.
(email: [email protected]; [email protected]).
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O
ur intuition tells us that stationary waves, which have a
constant intensity throughout an extended region of
space, can only exist when no obstacles hamper the
wave’s free propagation. Such an obstacle could be an electrostatic
potential for an electronic matter wave, the non-uniform
distribution of a dielectric medium for an electromagnetic wave
or a wall that reflects an acoustic pressure wave. All of these cases
lead to scattering, diffraction and wave interference, resulting in
the highly complex variation of a wave’s spatial profile that
continues to fascinate us in all its different manifestations.
Suppressing or merely controlling these effects, which are at the
heart of wave physics, is a challenging task, as the quest for a
cloaking device1 or the research in adaptive optics2, and in
wavefront shaping through complex media3 make us very much
aware. Strategies in this direction are thus in high demand and
would fall on a fertile ground in many of the different disciplines
of science and technology in which wave propagation is a key
element.
A new avenue to explore various wave phenomena has recently
been opened up when it was realized that waves give rise to very
unconventional features when being subject to a suitably chosen
spatial distribution of both gain and loss. Such non-Hermitian
potential regions4,5, which serve as sources and sinks for waves,
respectively, can give rise to novel wave effects that are impossible
to realize with conventional, Hermitian potentials. Examples of
this kind, which were meanwhile also realized in the
experiment6–10, are the unidirectional invisibility of a gain–loss
potential11, devices that can simultaneously act as laser and as a
perfect absorber12–14 and resonant structures with unusual
features like non-reciprocal light transmission10 or loss-induced
lasing15–17. In particular, systems with a so-called parity-time
(PT ) symmetry18, where gain and loss are carefully balanced,
have recently attracted enormous interest in the context of nonHermitian photonics19–24.
Inspired by these recent advances, we show here that for a
general class of potentials with gain and loss, it is possible to
construct constant-intensity wave solutions. Quite surprisingly,
these are solutions to both the paraxial equation of diffraction and
the nonlinear Schrödinger equation (NLSE). In the linear regime,
such constant-intensity waves resemble Bessel beams of free
space25. They carry infinite energy, but retain many of their
exciting properties when being truncated by a finite-size input
aperture. In the nonlinear regime, this class of waves turns out to
be of fundamental importance, as they provide the first instance to
investigate the best known symmetry breaking instability, that is,
the so-called modulational instability (MI)26–31, in inhomogeneous
potentials. Using these solutions for studying the phenomenon of
MI, we find that in the self-focusing case, unstable periodic modes
appear causing the wave to disintegrate and to generate a train of
complex solitons. In the defocusing regime, the uniform intensity
solution is modulationally unstable for some wavenumbers.
Results
One-dimensional constant-intensity waves. Our starting point is
the well-known NLSE. This scalar wave equation encompasses
many aspects of optical wave propagation as well as the physics of
matter waves. Specifically, we will consider the NLSE with a
general, non-Hermitian potential V(x) and a Kerr nonlinearity,
@c @ 2 c
þ 2 þ V ðxÞc þ g jcj2 c ¼ 0
ð1Þ
@z
@x
The scalar, complex valued function c(x,z) describes the
electric field envelope along a scaled propagation distance z or the
wave function of a matter wave as it evolves in time. The
nonlinearity can either be self-focusing or defocusing, depending
on the sign of g. For this general setting, we now investigate a
i
2
whole family of recently introduced potentials V(x) (ref. 32),
which are determined by the following relation,
V ðxÞ¼ W 2 ðxÞ i
dW ðxÞ
dx
ð2Þ
where W(x) is a given real function. In the special case where
W(x) is even, the actual optical potential V(x) turns out to be
PT -symmetric, since V(x) ¼ V*( x). We emphasize, however,
that our analysis is also valid for confined, periodic or disordered
potentials W(x), which do not necessarily lead to a
PT -symmetric form of V(x)
R þ 1 (but for which gain and loss are
always balanced since 1 Im½V ðxÞdx¼ 0 in the case of
localized
or
periodic
potentials.).
For the
entire
non-Hermitian family of potentials that are determined by
equation 2 (see Methods), we can prove that the following
analytical and stationary constant-intensity wave is a solution to
the NLSE in equation 1,
R
2
ð3Þ
cðx; zÞ ¼ A eigA z þ i WðxÞdx
notably with a constant and real amplitude A. We emphasize here
the remarkable fact that this family of solutions exists in the linear
regime (g ¼ 0) as well as for arbitrary strength of nonlinearity
(g ¼ ±1). Under linear conditions (g ¼ 0), the constant-intensity
wave given by equation 3 is one of the radiation eigenmodes (not
confined) of the potential with propagation constant equal to
zero. (non-zero propagation constants are obtained by adding a
constant term to the potential VðxÞ in equation 2). Another
interesting point to observe is that the above solutions exist only
for non-Hermitian potentials, since for W(x)-0 we also have
V(x)-0. Therefore, these families of counterintuitive solutions
are a direct consequence of the non-Hermitian nature of the
involved potential V(x) and as such exist only for these complex
structures with gain and loss. The fact that such constantintensity waves are a direct generalization of the fundamental
concept of free-space plane waves to complex environments can
be easily understood by setting W(x) ¼ c1 ¼ const. with c1 2 R: In
this case, the potential V(x) ¼ c21 corresponds to a bulk dielectric
medium for which the constant-intensity waves reduce to the
plane waves of homogeneous space, c ¼ eic1 x . It can also be
shown that the potential W(x) determines the power flow in the
transverse plane that physically forces the light to flow from the
gain to the loss regions. In particular, the transverse normalized
Poynting vector defined as S ¼ (i/2)(cqc*/qx c* qc/qx) takes
on the following very simple form: S ¼ A2W(x).
To illustrate the properties of such constant-intensity solutions,
we consider the following one-dimensional potentials (not
counting the direction of propagation z) generated
by Hermite
2
polynomials choosing W ðxÞ¼Hn ðxÞe Bx . The results for
vanishing non-linearity (g ¼ 0) and n ¼ 1, B ¼ 0.5 are shown in
Fig. 1. Note that the corresponding localized optical potential
V(x) is not PT -symmetric (Fig. 1a) and physically describes a
waveguide coupler with optical gain in the middle and lossy arms
in the evanescent region around it. If the initial beam is not
designed to have the correct phase (as given by equation 3) but is
instead c(x,0) ¼ A, then the light diffracts fast to the gain region,
as can be seen in Fig. 1b. In Fig. 1c,d, we show the results for the
constant-intensity solutions with the correct phase, where
diffraction is found to be strongly suppressed. Similar to the
diffraction-free beams25, we find that the wider the width of the
truncation aperture is at the input facet, z ¼ 0, the larger is the
propagation distance after which the beam starts to diffract
(compare Fig. 1c with Fig. 1d).
Two-dimensional constant-intensity waves. Similar constantintensity solutions can also be derived in two spatial dimensions
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a
b
5
||2
Potential
1
0
0
–1
40
–50
–2
0
x
20
0
2
50
x
c
z
d
1
||2
||2
1
0.5
0
0
40
–50
0
x
20
50
0.5
40
–50
z
x
0
20
50
z
Figure 1 | Constant-intensity waves in a linear waveguide coupler. (a) Real part (green line) and imaginary part (black line) of the complex potential V(x)
satisfying equation 2 (blue filled regions depict loss, whereas the red one depicts gain). (b) Evolution of a constant amplitude without the correct phase at
the input at z ¼ 0. (c,d) Spatial diffraction of the truncated constant-intensity solution satisfying the correct phase relation of equation 3. Two different input
truncations are shown for comparison. The lines in the x z planes of (b,c,d) around x ¼ 0 depict the real refractive index of the potential as shown in a.
Note the different vertical axis scale in b.
x,y. The family of these complex potentials V(x,y) and the
corresponding constant-intensity
solutions
c(x,y,z) of the
@2 c
@2 c
þ
þ
þ
V
ð
x;y
Þc þ g jcj2 c¼ 0
two-dimensional NLSE i @c
2
2
@z
@x
@y
are:
2
~ W
~
~ ir
ð4Þ
V ðx; yÞ¼W
~ W
~ ¼0
r
igA2 z þ i
ð5Þ
R
~ x
Wd~
ð6Þ
~
where W ~
x Wx þ~
y Wy with Wx,Wy being real functions of x,y
and C being any smooth open curve connecting an arbitrary
point (a,b) to any different point (x,y). As in the one-dimensional
case, these solutions are valid in both the linear and the nonlinear
domain. For the particular case of irrotational flow Wx ¼ cosx
siny, Wy ¼ cosy sinx, the resulting periodic potential V(x,y) is that
of an optical lattice with alternating gain and loss waveguides.
The imaginary part of such a lattice is shown in Fig. 2a. In Fig. 2b,
we display the diffraction of a constant-intensity beam with
the correct phase (as in equation 6) launched onto such a
linear lattice (g ¼ 0) through a circular aperture. As we can
see, the beam maintains its constant intensity over a remarkably
long distance. In Fig. 2c, we present the corresponding transverse
~
~ c rcÞ,
Poynting vector defined as ~
S ¼ ði=2Þðc rc
illustrating that the wave flux follows stream line patterns from
the gain to the loss regions. Once the finite beam starts to diffract,
this balanced flow is disturbed and the waves are concentrated in
the gain regions.
cðx;y;zÞ¼ Ae
C
Modulation instability of constant-intensity waves. Quite
remarkably, the above diffraction-free and uniform intensity
waves are also solutions of the NLSE for both the self-focusing
and the defocusing case. This allows us to study the modulation
instability of such solutions under small perturbations.
In particular, we are interested in understanding the linear
stability of the solutions of equation
1 of the form
il z iyðx;zÞ
e
cðx;zÞ ¼ A þ eFl ðxÞeilz þ eG
ð
x
Þe
, where the phase
R l
function is y(x,z) ¼ gA2z þ W(x) dx. This expression describes
the stationary constant-intensity wave under the perturbation of
o1. The imaginary
the eigenfunctions Fl(x) and Gl(x) with eo
part of l measures the instability growth rate of the perturbation
and determines whether a constant-intensity solution is
stable (l 2 R) or unstable (lAC). To leading order in e, we
obtain a linear eigenvalue problem for the two-component
perturbation eigenmodes ul(x)[Fl(x) Gl(x)]T, the eigenvalues
of which are l. This eigenvalue problem and the operator matrix
$
M are defined in the Methods section. So far the presented
MI-analysis is general and can be applied to any real W(x)
(periodic or not). To be more specific, we now apply this
analysis to study the MI of constant-intensity waves in
PT -symmetric optical lattices19,20, assuming that W(x) is a
periodic potential with period a. In particular, we consider
the example of a PT -symmetric photonic lattice where
W ðxÞ ¼ V20 þ V 1 cosðxÞ (the resulting optical potential and the
corresponding constant-intensity solution are given in the
Methods section). For all the subsequent results we will always
assume (without loss of generality) that V0 ¼ 4 and V1 ¼ 0.2. It is
important to note here that for these parameters our PT -lattice
V(x) is in the so-called ‘unbroken PT -symmetric phase’ with
only real propagation constants (see Methods). In the broken
phase, some of these eigenvalues are complex and the instabilities
due to nonlinearity are physically expected. As W(x) is periodic
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a
b
50
10
5
y
y
z
–50
8
0
0
–50
0
50
x
–5
6
–5
0
50
5
y
x
0
4
–50
c
–50
2
50
0
x
2
50
y
0
y
0
–50
–50
0
50 50 0
x
y
–2
–2
0
0
–50
–50
0
50
x
2
x
4
1
0.2
0.8
|Im (λ)|
0.25
0.15
0.1
0
0
0.6
0.4
0.2
0.05
0.2
0
0.4
0
0.4
0.2
k
k
0.06
0.08
0.06
0.04
|Im (λ)|
|Im (λ)|
we can expand the perturbation eigenvectors ul(x) in a Fourier
series and construct numerically the bandstructure of the stability
problem (different from the physical band-structure of the optical
lattice). Based on the above, the Floquet–Bloch theorem implies
that the eigenfunctions ul(x) can be written in the form
ul(x) ¼ /(x,k)eikx, where /(x,k) ¼ /(x þ a,k) with k being the
Bloch momentum of the stability problem (see Methods). The
corresponding results are illustrated in Fig. 3a,b for a self-focusing
nonlinearity (g ¼ 1) and for different values of the amplitude A.
More specifically, we show the instability growth rate |Im{l(k)}|
as a function of the Bloch wavenumber k of the perturbation
eigenvector in the first half Brillouin zone. We see that the
constant-intensity waves are linearly unstable for any value of the
Bloch momentum of the imposed perturbation and that
instability band gaps form due to the periodic nature of the
imposed perturbations. The different bands are illustrated in
Fig. 3a,b with different colours.
The situation is different for the defocusing case (g ¼ 1)
where the results are presented in Fig. 3c,d. For some values of k,
the constant-intensity solutions are linearly stable and their
instability dependence forms bands reminiscent of the bands
appearing in conventional MI results for bulk or periodic
potentials26,29,30, but quite different and profoundly more
complex.
To understand the physical consequences of such instabilities
and how they lead to filament formation, we have performed
independent numerical simulations for the dynamics of the
constant-intensity solutions against specific perturbations with
results being shown in Fig. 4. More specifically, we examine the
intensity evolution of a constant-intensity solution when
being perturbed by a specific Floquet–Bloch stability mode. In
other words, at the input
of the waveguide structure at z ¼ 0,
we have cðx;z¼ 0Þ ¼ A þ eFl ðxÞ þ eGl ðxÞ eiyðx;0Þ , with phase
yðxÞ¼ V20 x þ V1 sinðxÞ and we want to know whether the linear
stability analysis captures the exponential growth of the imposed
perturbations correctly. For the considered PT -symmetric lattice
with self-focusing nonlinearity, we examine the nonlinear
|Im (λ)|
Figure 2 | Constant-intensity waves in a two-dimensional linear optical lattice. (a) Imaginary part of the complex potential V(x,y) discussed in the text.
Red and blue regions correspond to gain and loss, respectively. (b) Iso-contour of the beam intensity launched onto the potential in a through a circular
aperture of radius B40l0, where l0 is the free space wavelength. Also shown are three transverse intensity plots (from bottom to top) at z ¼ 0, z ¼ 5,
z ¼ 10. (c) Transverse power flow pattern (indicated by arrows) of the beam at z ¼ 5.
0.02
0.04
0.02
0
0
0
0.2
0.4
k
0
0.2
0.4
k
Figure 3 | Modulation instability diagrams for self-focusing and
defocusing nonlinearity. Growth rate of the instability |Im{l(k)}| as a
function of the Bloch momentum (half of first Brillouin zone), for selffocusing nonlinearity and amplitudes (a) A ¼ 0.5, (b) A ¼ 1 and for
defocusing nonlinearity (c) A ¼ 1 and (d) A ¼ 2. Different colours in
a and b denote different instability bands.
dynamics of the constant-intensity solution and the result is
presented in Fig. 4a. For a perturbation eigenmode with Bloch
momentum k ¼ 0 and A ¼ 1, e ¼ 0.01, we can see from Fig. 3b
that Im{l(0)}B1. Therefore, we can estimate the growth for a
propagation distance of z ¼ 5 to be around |1 þ 0.01 e1 5|2B6.1,
which agrees very well with the dynamical simulation of
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a
b
4.05
6
⎜⎜2
⎜⎜2
4
4
2
3.95
–5
0
x
5
2
4
6
8
40
–20
0
x
z
20
10
20
z
Figure 4 | Perturbed constant-intensity waves propagating in nonlinear media. Numerical results for the intensity evolution of a constant-intensity wave
for (a) a self-focusing nonlinearity (g ¼ 1) with parameters k ¼ 0, A ¼ 1, e ¼ 0.01, and (b) for a defocusing nonlinearity (g ¼ 1) with parameters k ¼ 0.22,
A ¼ 2, e ¼ 0.001 The peak values are indicated on the vertical axes and match very well with the results of our perturbation analysis.
Fig. 4a. Similarly, for the defocusing nonlinearity (Fig. 4b), and
for parameters k ¼ 0.22 and A ¼ 2, e ¼ 0.001, we estimate
the growth for a propagation distance z ¼ 35 to be around
|2 þ 0.001 e0.046 35|2B4.02, which matches very well with the
numerical propagation result of Fig. 4b.
Discussion
Symmetry breaking instabilities belong to the most fundamental
concepts of nonlinear sciences. They lead to many rich
phenomena such as pattern formation, self-focusing and
filamentation just to name a few. The best known symmetry
breaking instability is the MI. In its simplest form, it accounts
for the break up of a uniform intensity state due to
the exponential growth of random perturbations under the
combined effect of dispersion/diffraction and nonlinearity. Most
of the early work on MI has been related to classical
hydrodynamics, plasma physics and nonlinear optics. Soon
thereafter, it was realized that the idea of MI is in fact universal
and could exist in other physical systems. For example, spatial
optics is one particular area that provides a fertile ground
where MI can be theoretically modelled (mainly within the
framework of the NLSE) and experimentally realized. Indeed,
temporal MI has been observed in optical fibres as well as its
spatial counterpart in nonlinear Kerr, quadratic, biased
photorefractive media with both coherent and partially coherent
beams and in discrete waveguide arrays26–31. Up to now, most
of the mathematical modelling of MI processes has been focused
on wave propagation in homogeneous nonlinear media, where
an exact constant-intensity solution for the underlying
governing (NLSE type) equations can be obtained. In this
context, inhomogeneities are considered problematic as they
provide severe conceptual limitations that hinder one from
constructing a constant-intensity solution, a necessary condition
to carry out the MI analysis. Several directions have been
proposed to bypass this limitation. They can be organized into
three distinct categories: (i) the tight binding approach, in which
case the NLSE equation in the presence of an external periodic
potential is replaced by its discrete counterpart that, in turn,
admits an exact plane wave solution (a discrete Floquet–Bloch
mode), (ii) MI of nonlinear Bloch modes and (iii) direct
numerical simulations using a broad beam as an initial
condition whose nonlinear evolution is monitored. However,
none of these alternatives amount to true MI.
We overcome such difficulties by introducing the above family
of constant-intensity waves, which exists in a general class of
complex optical potentials. These type of waves have constant
intensity over all space despite the presence of non-Hermitian
waveguide structures. They also remain valid for any sign of
Kerr nonlinearity and thus allow us to perform a modulational
stability analysis for non-homogeneous potentials. The most
appropriate context to study the MI of such solutions is that of
PT -symmetric optics6–11,14,19–22,24. We find that in the selffocusing regime, the waves are always unstable, whereas in the
defocusing regime the instability appears for specific values of
Bloch momenta. In both regimes (self-focusing, defocusing), the
constant-intensity solutions break up into filaments following a
complex nonlinear evolution pattern.
We expect that our predictions can be verified by combining
recent advances in shaping complex wave fronts3 with new
techniques to fabricate non-Hermitian scattering structures with
gain and loss7–10. As the precise combination of gain and loss in
the same device is challenging, we suggest using passive structures
with only loss in the first place. For such suitably designed passive
systems6, solutions exist that feature a pure exponential decay in
the presence of an inhomogeneous index distribution. This
exponential tail should be observable in the transmission intensity
as measured at the output facet of the system. Another possible
direction is that of considering evanescently coupled waveguide
systems. Using coupled mode theory one can analytically show
that our constant-intensity waves exist also in such discrete
systems with distributed gain and loss all over the waveguide
channels. In this case, the constant-intensity waves are not
radiation modes but rather supermodes of the coupled system.
With these simplifications an experimental demonstration of
our proposal should certainly be within reach of current
technology.
Methods
Constant-intensity solutions of the non-Hermitian NLSE. We prove here
analytically that stationary constant-intensity solutions of the NLSE exist for
a wide class of non-Hermitian optical potentials (which are not necessarily
PT -symmetric). We are looking for solutions of the NLSE of the form
c(x,z) ¼ f(x)exp(imz), where f(x) is the complex field profile and m the corresponding propagation constant, to be found. By substitution of this last relation
into equation 1, we get the following nonlinear equation mf þ fxx þ V(x)f þ g|f|2
f ¼ 0. We assume a solution of the form f(x) ¼ r(x)exp[iY(x)], with r(x),Y(x) real
functions of position x. Since V(x) ¼ VR(x) þ iVI(x), the last nonlinear equation can
be separated in real and imaginary parts. As a result we get the following two
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coupled equations for the real and the imaginary part of the complex potential,
respectively:
rxx þ VR ðxÞ Y2x m r þ gr3 ¼ 0
ð7Þ
Yxx r þ 2Yx rx þ rVI ¼ 0
ð8Þ
where YxdY/dx and rxdr/dx. By choosing VR(x) ¼ Y2x, and by solving
equation 8 to get VI ðxÞ ¼ Yxx 2Yx rrx , we can reduce the above system of
coupled nonlinear ordinary differential equations to only one, namely rxx m r þ
gr3 ¼ 0. If we assume now a constant amplitude solution, namely r(x) ¼ A ¼ const.,
we have the following general solution for any real-valued phase function Y(x):
c(x,z) ¼ A exp[iY(x) þ igA2z], where VR ðxÞ¼Y2x , and VI(x) ¼ Yxx. By setting
W(x)Yx(x), we can write the optical potential, for which the constant-intensity
solution exists, as V(x) ¼ W2(x) idW/dx, and the
R constant-intensity solution
itself reads as follows cðx; z Þ ¼ Aexp½igA2 z þ i W ðxÞdx. We can easily see
that in the special case where W(x) is even, the actual optical potential V(x) is
PT -symmetric.
Modulation instability analysis in optical potentials. To study the modulation
instability of the uniform intensity states for any given W, we consider
small
perturbation of
the solutions of the NLSE of Rthe form: cðx; z Þ¼ A þ eFl ðxÞeilz
2
il z iyðx;z Þ
þ eGl ðxÞe
e
, where y(x,z) ¼ gA z þ W(x)dx and eo
o1. Here, Fl(x) and
Gl(x) are the perturbation eigenfunctions and the imaginary part of
l measures the instability growth rate of the perturbation. By defining the
perturbation two-component eigenmode ul(x)[Fl(x) Gl(x)]T, we obtain the
following linear eigenvalue problem (to leading order in e):
$
^ wl ðxÞ¼l wl ðxÞ
M L
ð9Þ
$
^ is defined by the following expression
where the operator matrix M L
$
^þ
L
gA2
^ ¼
ð10Þ
M L
2
^
gA
L
2
2
^0 ¼ gA þ d =dx
L
þ1
X
W n eiqnx
ð17Þ
n¼ 1
where q ¼ 2p/a is the dual lattice spacing. Substitution of equations 16 and 17 into
the eigenvalue problem of equation 9, leads us to the following nonlocal system of
coupled linear eigenvalue equations for the perturbation coefficients un,un and the
band eigenvalue l(k) that depends on the Bloch momentum k:
On ðkÞun þ1
X
Un;m ðkÞun-m þ gA2 un ¼ lun
m¼ 1
gA2 un On ðkÞun þ1
X
ð18Þ
U n;m ðkÞun-m ¼lun
m¼ 1
where Un,m(k) ¼ 2[q(n m) þ k]Wm and On(k) ¼ gA2 (qn þ k)2. The family of
constant-intensity wave solutions of the NLSE is modulationally unstable if there
exists a wave number k for which Im{l(k)}a0, while they are stable if l(k) is real.
For our case, the periodic function W is given by W ðxÞ ¼ V20 þ V 1 cosðqxÞ for
which equation 18 becomes:
mn ðkÞun an 1 ðkÞun 1 an þ 1 ðkÞun þ 1 þ gA2 un ¼ lun
gA2 un nn ðkÞun an 1 ðkÞun 1 an þ 1 ðkÞun þ 1 ¼ lun
ð19Þ
ð11Þ
2
ð12Þ
^1 ¼ 2W ðxÞd=dx
L
ð13Þ
So far the above discussion is general and applies to any (periodic or not) potential
W(x) that is real.
Properties of the PT -symmetric optical lattice. We choose a specific
example of a well-known non-Hermitian potential, that is, that of a
PT -symmetric optical lattice19,20. More specifically for the particular case of
W ðxÞ ¼ V20 þ V 1 cosðxÞ, we get the corresponding optical potential and constantintensity wave:
2
V0
V ðx Þ ¼
þ V 21 cos2 x þ V 0 V 1 cosx þ iV 1 sinx
ð14Þ
4
V0x
cðx; z Þ¼ Aexp igA2 z þ i
þ iV 1 sinx
2
ð15Þ
It is obvious that this potential is PT -symmetric as it satisfies the symmetry
relation V(x) ¼ V*( x). In order for the constant-intensity solution to be periodic
in x with the same period as the lattice, the term V0 must be quantized,
namely V0 ¼ 0,±2,±4,.... This constant term that appears in the potential
W(x) results in another constant term in the actual potential V(x) and can be
removed (with respect to the NLSE) with a gauge transformation of the type
~ ðx; z ÞeizV 20 =4 . Even though this is the case, this term is important because
cðx; z Þ¼c
it also appears in the real part of V(x). It determines if the PT -lattice is in the
broken or in the unbroken phase, regarding its eigenspectrum. For the considered
parameters, the lattice is below the exceptional point and its eigenvalue
spectrum is real.
Plane wave expansion method. Even though our methodology is general,
we apply it to study the modulation instability of constant-intensity waves in
PT -symmetric optical lattices. In particular, we consider the periodic W(x)
(with period a) that leads to equations 14 and 15. As we are interested in the
MIs of the constant-intensity wave solution of the NLSE under self-focusing and
defocusing nonlinearities, we want the PT -lattice V(x) to be in the unbroken
phase. In the broken phase some eigenvalues are complex and the instabilities
are physically expected. That is the reason why we choose (without loss of
generality) the parameters V0 ¼ 4 and V1 ¼ 0.2, which lead to an ‘unbroken’
spectrum with real eigenvalues. As W(x) is periodic, we can expand the
perturbation eigenvectors ul(x) in Fourier series and construct numerically
the band-structure of the stability problem. So at this point, we have to distinguish
6
W ðxÞ¼
where an(k) ¼ V1(qn þ k), mn(k) ¼ On(k) V0(qn þ k) and
nn(k) ¼ On(k) þ V0(qn þ k).
and the related linear operators are defined by the relationships:
^0 iL
^1
^ ¼ L
L
between the physical band-structure of the problem and the perturbation
band-structure of the stability problem of equation 9. Based on the above,
the Floquet–Bloch theorem implies that the eigenfunctions ul(x) can be written in
the form ul(x) ¼ /(x,k)eikx, where /(x,k) ¼ /(x þ a,k) with k being the Bloch
momentum of the stability problem. Applying the plane wave expansion
method, the wavefunctions /(x,k) and the potential W(x) can be expanded in
Fourier series as:
þ1 X
un ðkÞ iðqn þ kÞx
wl ðxÞ¼
ð16Þ
e
u n ðkÞ
n¼ 1
Direct eigenvalue method. An alternative way (instead of the plane wave
expansion method that was used above) of solving the infinite dimensional
eigenvalue problem of equation 9 is to directly apply the Floquet–Bloch theorem on
the eigenfunctions ul(x), employ the Born-von-Karman boundary conditions
(periodic boundary conditions at the end points of the finite lattice) and construct
numerically the bandstructure of the instability growth for every value of the
Bloch momentum. In particular, the eigenfunctions can be written as
ul(x) ¼ [u(x)eikx u(x)eikx]T, where u(x) ¼ u(x þ a), u(x) ¼ u(x þ a). Substituting
this form of the perturbation eigenfunctions into equation 9, we get the
following eigenvalue problem:
!
^ þ þ 2ikd=dx k2 2kW ðxÞ
L
gA2
^ 2ikd=dx þ k2 2kW ðxÞ
gA2
L
ð20Þ
u
u
¼ lð k Þ
u
u
where the Bloch momentum takes values in the first Brillouin zone
^þ ; L
^ are defined by equations 11–13.
kA[ p/a,p/a] and the operators L
By applying the finite difference method, we restrict our analysis to one unit cell
xA[ a/2,a/2], in order to calculate the growth rate of the random perturbations
for every value of the Bloch momentum. We have checked explicitly that both
approaches, that is, the plane wave expansion method based on equation 19 and
the direct eigenvalue analysis based on equation 20, give the same results.
Analytical results in the shallow lattice limit. In the limit of a shallow optical
lattice (the refractive index difference between the periodic modulation and the
background refractive index value is very small), one can gain substantial insight
into the structure of the unstable band eigenvalues by deriving an approximate
analytical expression for l(k) valid near the Bragg points based on perturbation
theory. These points are given by (for self-focusing nonlinearity):
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
ln þ ðkÞ¼ ðk þ nqÞ2 ðk þ nqÞ2 kc
ð21Þ
ln ðkÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðknqÞ2 ðk nqÞ2 kc
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l0 ðkÞ¼ k2 k2 kc
ð22Þ
ð23Þ
¼ 2A2
and n ¼ 1,2,3,.... The above analytical formulas lead to an excellent
where kc
match with the numerical approaches in the shallow lattice limit (V0, V1oo1).
Complex filament formation. To understand better the complex filament
formation of a constant-intensity solution in a PT -symmetric lattice for both signs
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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms8257
of nonlinearity, we performed nonlinear wave propagation simulations based on a
spectral fast Fourier approach of the integrating factors method for NLSE. The
initial conditions that were used to examine the filament formation were based on
the perturbation eigenmode profiles. In particular, we have at z ¼ 0, the following
initial
eigenfunctions uðxÞ; uðxÞ : cðx; z ¼ 0Þ
field profile in terms of Bloch
¼ A þ euk ðxÞeikx þ euk ðxÞe ikx eiyðxÞ , for specific values of Bloch momentum k
and the constant-intensity wave amplitude A.
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Acknowledgements
K.G.M. is supported by the People Programme (Marie Curie Actions) of the European
Union’s Seventh Framework Programme (FP7/2007–2013) under REA grant agreement
number PIOF-GA-2011–303228 (project NOLACOME). Z.H.M. was supported in part
by NSF Grant No. DMS-0908599. The work of D.N.C. was partially supported by NSF
Grant No. ECCS-1128520 and the Air Force Office of Scientific Research Grants Nos.
FA9550-12-1-0148 and FA9550-14-1-0037. S.R. acknowledges financial support by the
Austrian Science Fund (FWF) through Project SFB NextLite (F49-P10) and Project
GePartWave (I1142).
Author contributions
All authors have contributed to the development and/or implementation of the concept,
discussed and analysed the results. K.G.M. and Z.M. carried out the analytical and
numerical calculations. S.R. and D.C. provided theoretical and conceptual support.
K.G.M., Z.M. and S.R. wrote the manuscript with input from all authors.
Additional information
Competing financial interests: The authors declare no competing financial interests.
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How to cite this article: Makris, K. G. et al. Constant-intensity waves and their
modulation instability in non-Hermitian potentials. Nat. Commun. 6:7257
doi: 10.1038/ncomms8257 (2015).
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